Clifford Algebra-Valued Research Articles

Clifford Algebra-Valued Segal–Bargmann Transform and Taylor Isomorphism

Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $$\mathbb {R}^n$$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $$\mathbb {R}^{n+1}$$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of $$\mathbb {R}^n$$ .

Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Helmholtz Operator

Radiation Conditions and Integral Representations for Clifford Algebra-Valued Null-Solutions of the Helmholtz Operator

On the Zeroes of Clifford Algebra-Valued Polynomials with Paravector Coefficients

In this paper, we first show the zeros of monic Clifford algebra polynomials with paravector coefficients by generalizing a right-division algorithm for quaternion polynomials. Then we provide a necessary and sufficient condition for the polynomials, which have an infinite number of roots. In particular, we derive the explicit formulas for computing the zeros of monic Clifford algebra quadratic polynomials with paravector coefficients and some cubic, quartic polynomials with real coefficients. From this, we obtain that any monic Clifford algebra quadratic polynomial with paravector coefficients has at least one zero.

Three-Term Recurrence Relations for Systems of Clifford Algebra-Valued Orthogonal Polynomials

This work was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications of the University of Aveiro, the CMAT - Research Centre of Mathematics of the University of Minho and the FCT - Portuguese Foundation for Science and Technology (“Fundacao para a Ciˆencia e a Tecnologia”), within projects PEst-OE/MAT/UI4106/2014 and PEst-OE/MAT/UI0013/2014.

A Matrix Recurrence for Systems of Clifford Algebra-Valued Orthogonal Polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory (Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Gürlebeck, Bock, Lávička, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.

The Szegö Metric Associated to Hardy Spaces of Clifford Algebra Valued Functions and Some Geometric Properties

In analogy to complex function theory we introduce a Szego metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in R m+1 . These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szego metric turns out to have a pseudo-invariance under Mobius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.

Fundamental Theorems for Clifford Algebra-Valued Distributions in Elliptic Clifford Analysis

A fundamental theorem in Elliptic Clifford Analysis (ECA), with the standard vector Dirac operator, is presented that is valid for Clifford algebra-valued distributions. This theorem holds under fairly general conditions on the allowed singularities of the right-hand side distributions and on the region of integration. Next a specialization of this fundamental theorem is proved that forms the starting point for solving boundary value problems with distributional sources in ECA. Finally, distributional equivalents of the Residue theorem, Cauchy’s theorem and Cauchy’s integral theorem are stated.

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cln,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \({\mathbb{R}^{n}}\) . We express the admissibility condition in terms of the Cln,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cln,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.

Partial Primitives for Clifford Algebra-Valued Functions

In this paper we generalize the concept of primitivation of monogenic functions taking values in a Clifford algebra, which is on its own a generalization to higher dimension of the primitivation problem for holomorphic functions in the complex plane. This problem can be stated as follows: given a monogenic function \(f(x_0, \underline{x})\) on \({\mathbb{R}}^{m+1}\) , i.e. a solution for the generalized Cauchy-Riemann operator D on \({\mathbb{R}}^{m+1}\) , construct a monogenic function \(g(x_0, \underline{x})\) such that \(\overline{D}g = f\). In view of the fact that, for monogenic functions g, this can be written as \(\partial_{x_0}\)g = f, a straightforward generalization consists in replacing the scalar generator \(\partial_{x_0}\) of translations in the x0-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.

Editorial

Editorial